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Question Number 20934 Answers: 0 Comments: 1

If z satisfies ∣z − 1∣ < ∣z + 3∣, then ω = 2z + 3 − i satisfies (1) ∣ω − 5 − i∣ < ∣ω + 3 + i∣ (2) ∣ω − 5∣ < ∣ω + 3∣ (3) Im (iω) > 1 (4) ∣arg(ω − 1)∣ < (π/2)

$$\mathrm{If}\:{z}\:\mathrm{satisfies}\:\mid{z}\:−\:\mathrm{1}\mid\:<\:\mid{z}\:+\:\mathrm{3}\mid,\:\mathrm{then}\:\omega\:= \\ $$$$\mathrm{2}{z}\:+\:\mathrm{3}\:−\:{i}\:\mathrm{satisfies} \\ $$$$\left(\mathrm{1}\right)\:\mid\omega\:−\:\mathrm{5}\:−\:{i}\mid\:<\:\mid\omega\:+\:\mathrm{3}\:+\:{i}\mid \\ $$$$\left(\mathrm{2}\right)\:\mid\omega\:−\:\mathrm{5}\mid\:<\:\mid\omega\:+\:\mathrm{3}\mid \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Im}\:\left({i}\omega\right)\:>\:\mathrm{1} \\ $$$$\left(\mathrm{4}\right)\:\mid\mathrm{arg}\left(\omega\:−\:\mathrm{1}\right)\mid\:<\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 20933 Answers: 1 Comments: 0

If z is a complex number satisfying z + z^(−1) = 1, then z^n + z^(−n) , n ∈ N, has the value (1) 2(−1)^n , when n is a multiple of 3 (2) (−1)^(n−1) , when n is not a multiple of 3 (3) (−1)^(n+1) , when n is a multiple of 3 (4) 0 when n is not a multiple of 3

$$\mathrm{If}\:{z}\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number}\:\mathrm{satisfying} \\ $$$${z}\:+\:{z}^{−\mathrm{1}} \:=\:\mathrm{1},\:\mathrm{then}\:{z}^{{n}} \:+\:{z}^{−{n}} ,\:{n}\:\in\:{N},\:\mathrm{has} \\ $$$$\mathrm{the}\:\mathrm{value} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{2}\left(−\mathrm{1}\right)^{{n}} ,\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3} \\ $$$$\left(\mathrm{2}\right)\:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} ,\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of} \\ $$$$\mathrm{3} \\ $$$$\left(\mathrm{3}\right)\:\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} ,\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{0}\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3} \\ $$

Question Number 20932 Answers: 0 Comments: 0

If a, b, c are real numbers and z is a complex number such that, a^2 + b^2 + c^2 = 1 and b + ic = (1 + a)z, then ((1 + iz)/(1 − iz)) equals. (1) ((b − ic)/(1 − ia)) (2) ((a + ib)/(1 + c)) (3) ((1 − c)/(a − ib)) (4) ((1 + a)/(b + ic))

$$\mathrm{If}\:{a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{and}\:{z}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{complex}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that},\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \\ $$$$=\:\mathrm{1}\:\mathrm{and}\:{b}\:+\:{ic}\:=\:\left(\mathrm{1}\:+\:{a}\right){z},\:\mathrm{then}\:\frac{\mathrm{1}\:+\:{iz}}{\mathrm{1}\:−\:{iz}} \\ $$$$\mathrm{equals}. \\ $$$$\left(\mathrm{1}\right)\:\frac{{b}\:−\:{ic}}{\mathrm{1}\:−\:{ia}} \\ $$$$\left(\mathrm{2}\right)\:\frac{{a}\:+\:{ib}}{\mathrm{1}\:+\:{c}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{1}\:−\:{c}}{{a}\:−\:{ib}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}\:+\:{a}}{{b}\:+\:{ic}} \\ $$

Question Number 20926 Answers: 0 Comments: 0

Question Number 20925 Answers: 2 Comments: 0

In a △ABC, if a=2, b=60° and c=75°, then b =

$$\mathrm{In}\:\mathrm{a}\:\bigtriangleup{ABC},\:\mathrm{if}\:{a}=\mathrm{2},\:{b}=\mathrm{60}°\:\mathrm{and}\:{c}=\mathrm{75}°, \\ $$$$\mathrm{then}\:{b}\:= \\ $$

Question Number 20927 Answers: 1 Comments: 0

(a+b)×(a+b)

$$\left({a}+{b}\right)×\left({a}+{b}\right) \\ $$

Question Number 20916 Answers: 1 Comments: 0

A body starts rotating about a stationary axis with an angular acceleration b = 2t rad/s^2 . How soon after the beginning of rotation will the total acceleration vector of an arbitrary point on the body forms an angle of 60° with its velocity vector? (1) (2(√3))^(1/3) s (2) (2(√3))^(1/2) s (3) (2(√3)) s (4) (2(√3))^2 s

$$\mathrm{A}\:\mathrm{body}\:\mathrm{starts}\:\mathrm{rotating}\:\mathrm{about}\:\mathrm{a} \\ $$$$\mathrm{stationary}\:\mathrm{axis}\:\mathrm{with}\:\mathrm{an}\:\mathrm{angular} \\ $$$$\mathrm{acceleration}\:{b}\:=\:\mathrm{2}{t}\:\mathrm{rad}/\mathrm{s}^{\mathrm{2}} .\:\mathrm{How}\:\mathrm{soon} \\ $$$$\mathrm{after}\:\mathrm{the}\:\mathrm{beginning}\:\mathrm{of}\:\mathrm{rotation}\:\mathrm{will}\:\mathrm{the} \\ $$$$\mathrm{total}\:\mathrm{acceleration}\:\mathrm{vector}\:\mathrm{of}\:\mathrm{an}\:\mathrm{arbitrary} \\ $$$$\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{body}\:\mathrm{forms}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{60}° \\ $$$$\mathrm{with}\:\mathrm{its}\:\mathrm{velocity}\:\mathrm{vector}? \\ $$$$\left(\mathrm{1}\right)\:\left(\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{3}} \:\mathrm{s} \\ $$$$\left(\mathrm{2}\right)\:\left(\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{2}} \:\mathrm{s} \\ $$$$\left(\mathrm{3}\right)\:\left(\mathrm{2}\sqrt{\mathrm{3}}\right)\:\mathrm{s} \\ $$$$\left(\mathrm{4}\right)\:\left(\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \:\mathrm{s} \\ $$

Question Number 20915 Answers: 1 Comments: 0

Two shells are fired from a canon with speed u each, at angles of α and β respectively with the horizontal. The time interval between the shots is t. They collide in mid air after time T from the first shot. Which of the following conditions must be satisfied? (a) α > β (b) T cos α = (T − t) cos β (c) (T − t) cos α = T cos β (d) u sin α T − (1/2) g T^2 = u sin β (T − t) − (1/2) g (T − t)^2

$$\mathrm{Two}\:\mathrm{shells}\:\mathrm{are}\:\mathrm{fired}\:\mathrm{from}\:\mathrm{a}\:\mathrm{canon}\:\mathrm{with}\:\mathrm{speed}\:\mathrm{u}\:\mathrm{each},\:\mathrm{at} \\ $$$$\mathrm{angles}\:\mathrm{of}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{respectively}\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{The} \\ $$$$\mathrm{time}\:\mathrm{interval}\:\mathrm{between}\:\mathrm{the}\:\mathrm{shots}\:\mathrm{is}\:{t}.\:\mathrm{They}\:\mathrm{collide}\:\mathrm{in}\:\mathrm{mid} \\ $$$$\mathrm{air}\:\mathrm{after}\:\mathrm{time}\:{T}\:\mathrm{from}\:\mathrm{the}\:\mathrm{first}\:\mathrm{shot}.\:\mathrm{Which}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{conditions}\:\mathrm{must}\:\mathrm{be}\:\mathrm{satisfied}? \\ $$$$\left({a}\right)\:\alpha\:>\:\beta \\ $$$$\left({b}\right)\:{T}\:\mathrm{cos}\:\alpha\:=\:\left({T}\:−\:{t}\right)\:\mathrm{cos}\:\beta \\ $$$$\left({c}\right)\:\left({T}\:−\:{t}\right)\:\mathrm{cos}\:\alpha\:=\:{T}\:\mathrm{cos}\:\beta \\ $$$$\left({d}\right)\:{u}\:\mathrm{sin}\:\alpha\:{T}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\:{g}\:{T}^{\mathrm{2}} \:=\:{u}\:\mathrm{sin}\:\beta\:\left({T}\:−\:{t}\right)\:−\:\frac{\mathrm{1}}{\mathrm{2}}\:{g}\:\left({T}\:−\:{t}\right)^{\mathrm{2}} \\ $$

Question Number 20914 Answers: 1 Comments: 0

The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has (1) Only purely imaginary roots (2) All real roots (3) Two real and two purely imaginary roots (4) Neither real nor purely imaginary roots

$$\mathrm{The}\:\mathrm{quadratic}\:\mathrm{equation}\:{p}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{with} \\ $$$$\mathrm{real}\:\mathrm{coefficients}\:\mathrm{has}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\mathrm{roots}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{equation}\:{p}\left({p}\left({x}\right)\right)\:=\:\mathrm{0} \\ $$$$\mathrm{has} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Only}\:\mathrm{purely}\:\mathrm{imaginary}\:\mathrm{roots} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{All}\:\mathrm{real}\:\mathrm{roots} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Two}\:\mathrm{real}\:\mathrm{and}\:\mathrm{two}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\mathrm{roots} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Neither}\:\mathrm{real}\:\mathrm{nor}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\mathrm{roots} \\ $$

Question Number 20908 Answers: 1 Comments: 0

integrate with respect to x ∫(((2x+1)/(x^2 +4x+8)))dx

$${integrate}\:{with}\:{respect}\:{to}\:{x}\: \\ $$$$\int\left(\frac{\mathrm{2}{x}+\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{8}}\right){dx} \\ $$

Question Number 20907 Answers: 1 Comments: 0

How many zeroes (0) there in 1×2×3×4..........99×100

$${How}\:{many}\:{zeroes}\:\left(\mathrm{0}\right)\:{there}\:{in}\:\mathrm{1}×\mathrm{2}×\mathrm{3}×\mathrm{4}..........\mathrm{99}×\mathrm{100}\: \\ $$

Question Number 20905 Answers: 1 Comments: 1

∫e^x^2 dx

$$\int{e}^{{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 20903 Answers: 1 Comments: 0

Question Number 20891 Answers: 0 Comments: 11

The Figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed ω and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed ω/2. The ring and disc are separated by frictionless ball bearing. The system is in the x-z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle 30° with the horizontal. Then with respect to the horizontal surface (a) The point O has a linear velocity 3Rωi^∧ (b) The point P has a linear velocity ((11)/4)Rωi^∧ + ((√3)/4)Rωk^∧

$$\mathrm{The}\:\mathrm{Figure}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{system}\:\mathrm{consisting} \\ $$$$\mathrm{of}\:\left({i}\right)\:\mathrm{a}\:\mathrm{ring}\:\mathrm{of}\:\mathrm{outer}\:\mathrm{radius}\:\mathrm{3}{R}\:\mathrm{rolling} \\ $$$$\mathrm{clockwise}\:\mathrm{without}\:\mathrm{slipping}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{surface}\:\mathrm{with}\:\mathrm{angular}\:\mathrm{speed} \\ $$$$\omega\:\mathrm{and}\:\left({ii}\right)\:\mathrm{an}\:\mathrm{inner}\:\mathrm{disc}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{2}{R} \\ $$$$\mathrm{rotating}\:\mathrm{anti}-\mathrm{clockwise}\:\mathrm{with}\:\mathrm{angular} \\ $$$$\mathrm{speed}\:\omega/\mathrm{2}.\:\mathrm{The}\:\mathrm{ring}\:\mathrm{and}\:\mathrm{disc}\:\mathrm{are} \\ $$$$\mathrm{separated}\:\mathrm{by}\:\mathrm{frictionless}\:\mathrm{ball}\:\mathrm{bearing}. \\ $$$$\mathrm{The}\:\mathrm{system}\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:{x}-{z}\:\mathrm{plane}.\:\mathrm{The} \\ $$$$\mathrm{point}\:{P}\:\mathrm{on}\:\mathrm{the}\:\mathrm{inner}\:\mathrm{disc}\:\mathrm{is}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance} \\ $$$${R}\:\mathrm{from}\:\mathrm{the}\:\mathrm{origin},\:\mathrm{where}\:{OP}\:\mathrm{makes}\:\mathrm{an} \\ $$$$\mathrm{angle}\:\mathrm{30}°\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{Then} \\ $$$$\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{surface} \\ $$$$\left({a}\right)\:\mathrm{The}\:\mathrm{point}\:{O}\:\mathrm{has}\:\mathrm{a}\:\mathrm{linear}\:\mathrm{velocity} \\ $$$$\mathrm{3}{R}\omega\overset{\wedge} {{i}} \\ $$$$\left({b}\right)\:\mathrm{The}\:\mathrm{point}\:{P}\:\mathrm{has}\:\mathrm{a}\:\mathrm{linear}\:\mathrm{velocity} \\ $$$$\frac{\mathrm{11}}{\mathrm{4}}{R}\omega\overset{\wedge} {{i}}\:+\:\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}{R}\omega\overset{\wedge} {{k}} \\ $$

Question Number 20886 Answers: 1 Comments: 0

if sin x=msin y so proof that tan (1/2)(x−y)=((m−1)/(m+1))tan (1/2)(x+y)

$${if}\:\mathrm{sin}\:{x}={m}\mathrm{sin}\:{y} \\ $$$${so}\:{proof}\:{that} \\ $$$$\mathrm{tan}\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}−{y}\right)=\frac{{m}−\mathrm{1}}{{m}+\mathrm{1}}\mathrm{tan}\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}+{y}\right) \\ $$

Question Number 20885 Answers: 0 Comments: 0

if (θ−ϕ)subtle and sin θ+sin ϕ= (cos ϕ−cos θ)(√3) so proof sin 3θ+sin 3ϕ=0

$${if}\:\left(\theta−\varphi\right){subtle}\:{and}\:\:\:\mathrm{sin}\:\theta+\mathrm{sin}\:\varphi= \\ $$$$\left(\mathrm{cos}\:\varphi−\mathrm{cos}\:\theta\right)\sqrt{\mathrm{3}} \\ $$$${so}\:{proof}\:\mathrm{sin}\:\mathrm{3}\theta+\mathrm{sin}\:\mathrm{3}\varphi=\mathrm{0} \\ $$

Question Number 20884 Answers: 1 Comments: 0

2cos (π/3)cos ((9π)/(13))+cos ((3π)/(13))+cos ((5π)/(13))=0

$$\mathrm{2cos}\:\frac{\pi}{\mathrm{3}}\mathrm{cos}\:\frac{\mathrm{9}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{13}}=\mathrm{0} \\ $$

Question Number 20882 Answers: 1 Comments: 0